Integrand size = 29, antiderivative size = 29 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Int}\left (\frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}},x\right ) \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx \\ \end{align*}
Not integrable
Time = 87.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx \]
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Not integrable
Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86
\[\int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {2}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {8}{3}}}d x\]
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Timed out. \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {8}{3}}} \,d x } \]
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Not integrable
Time = 5.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {8}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Hanged} \]
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